All Questions
Tagged with isometries fa.functional-analysis
7 questions
14
votes
1
answer
925
views
What are the applications of the Mazur-Ulam Theorem?
Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
- 60.6k
11
votes
1
answer
339
views
What is an example of two Banach spaces $X,Y$ such that $X$ embeds isometrically but not linearly into $Y$?
By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.
Is this result known to fail for ...
8
votes
2
answers
602
views
If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$?
Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\...
- 623
6
votes
1
answer
1k
views
Must a surjective isometry on a dual space have a pre-adjoint?
Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^*$ is a surjective isometry of $X^*$.
In general, a linear map $g:X^* \...
- 63
4
votes
2
answers
341
views
A space isometric to $\ell_\infty^2$
Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$
Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...
- 1,879
2
votes
1
answer
602
views
Separable Banach spaces isometric to quotient of a Banach space
We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
- 585
1
vote
1
answer
300
views
Isometries of Hilbert space
It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two ...
- 1,361